BOSTON — The Cleveland Indians are not used to this. Their past is tortured, their present tattered. They have waited 68 years for a World Series title. Their chances seemed grim before this division series with the brawny Boston Red Sox.

But there they were on Monday night, bouncing around the grass at Fenway Park, bound for the American League Championship Series on Friday against the Toronto Blue Jays. The Indians’ tale of woe should have ended in misery. Now they have a shot at the World Series, after a three-game sweep.

They did it without a star outfielder, Michael Brantley, who barely played this season because of shoulder surgery. They did it without a top catcher, Jonathan Lucroy, who rejected a trade to them in July. They did it without two of their best starters, Danny Salazar, who hurt his forearm, and Carlos Carrasco, who broke a finger.

“This team’s been resilient from the start,” General Manager Mike Chernoff said. “Nobody built up the anxiety or built up the expectations. Guys just stepped in and took advantage of the opportunities.”

The last time the Indians took a playoff series, in 2007, they won the final postseason game at the old Yankee Stadium and drove Joe Torre — a four-time champion manager — away from the Bronx. On Monday, the Indians dropped the curtain on the David Ortiz Show, the longest-running, most successful New England baseball act in about a century.

The Red Sox smoked blistering liners as they tried to mount a rally in the eighth. They threatened again in the ninth, trying desperately to extend the series and Ortiz’s career. Both times, Cody Allen worked out of trouble, stranding four runners in all. Travis Shaw flied to right to end it.

“Every team that makes it to the playoffs believes in themselves,” said Coco Crisp, a midseason acquisition whose two-run homer helped the Indians steal a 4-3 victory. “As soon as I came in, I felt the chemistry. Everybody loves each other. It’s almost like a family. These guys grind and they believe they’re going to get it done.”

Playing for the Red Sox in 2007, Crisp caught the final out of the A.L.C.S. to beat the Indians and complete a comeback from a three-games-to-one deficit. The Indians gained a measure of revenge on Monday, when their depleted rotation again managed just fine.

Trevor Bauer wobbled a bit in Game 1, but Corey Kluber overwhelmed the Red Sox in Game 2 and Josh Tomlin allowed just four hits and two runs in five innings on Monday.

Boston’s Rick Porcello and David Price, who combined to earn $50 million this season, lost their starts in this series. Kluber — an ordinary Class AA prospect when he arrived six years ago in a trade for Jake Westbrook – won his, as did Tomlin, a former 19th-round draft pick with a 4.58 career earned run average.

The crowd chanted his name to try to intimidate Tomlin, but it had the opposite effect.

“I was surprised there was that many people that knew my name, to be honest with you,” Tomlin said. “It was awesome. I can honest to God tell you it was hard for me to hear anything for the first couple of innings, and once they started chanting my name, it kind of became real. I knew where I was at.”

Terry Francona, the Indians manager who guided the Red Sox to championships in 2004 and 2007, got everything he could have wanted from Tomlin, and turned the game over to Andrew Miller after a leadoff single in the sixth.

Miller reported to spring training with the Yankees in February and immediately stifled any controversy about ceding the closer spot to Aroldis Chapman. Miller said his job was to get outs whenever the manager asked — a credo he carried with him to Cleveland, when the Yankees traded him there in July.

Unburdened by strict bullpen roles, Francona has used Miller as something of an old-time pitching fireman, snuffing rallies before they get out of control. He called for him in the fifth inning of Game 1, and Miller held the lead into the seventh. After three days of rest, Miller worked two scoreless innings on Monday.

“The momentum in these games is so crazy how fast it can swing,” Miller said. “I think I’m just ready for anything. I think we all are out there.”

Miller, Brian Shaw and Allen ushered the Red Sox off the stage, just as Roberto Osuna and the Blue Jays’ bullpen did on Sunday to the Texas Rangers, completing another three-game sweep. That can happen easily in baseball, a sport in which even the worst team wins about 60 games in a season. But a quick dismissal is still jarring, and Boston has felt it before.

Somewhat incredibly, this was the seventh time in eight seasons that the Red Sox did not win a postseason game. They were swept in this round by the Los Angeles Angels in 2009 but won the World Series in 2013.

Their front office leadership has changed twice in that time, from Theo Epstein to Ben Cherington to Dave Dombrowski. They have also changed managers twice, from Francona to Bobby Valentine to Farrell, whose job status could come under scrutiny in a town that does not tolerate losing.

Cleveland loses more often, of course, but the Indians have a sound enough process that they have become an industry model. Several teams have hired Cleveland executives to lead their baseball operations, including the Pittsburgh Pirates (Neal Huntington), the Minnesota Twins (Derek Falvey) and the Blue Jays, whose president, Mark Shapiro, and general manager, Ross Atkins, both came from the Indians.

The Indians’ opening-day payroll, about $96 million, was more than $100 million less than Boston’s. But Cleveland is celebrating, and the Red Sox are done. “Their hands are tied in what they’re able to do,” said Mike Napoli, the veteran first baseman. “But to be able to get the right guys in here to become leaders and show guys how to play the game and come together as a group, they did a great job. We all did it together.”

A version of this article appears in print on , on Page B9 of the New York edition with the headline: Team Rewrites Hard-Luck Tale and Moves On. Order Reprints | Today’s Paper | Subscribe